Three-State Quantum Clock Model

• The three-state quantum clock model is a discrete Z3 lattice framework with three-level clock variables that exhibits a self-dual second-order quantum critical point at (h/J)=1.
• Its critical dynamics are characterized by equilibrium scaling, with correlation-length exponent ν₃≈0.842 and Loschmidt echo minima following a finite-size scaling law.
• Chiral deformations and antiferromagnetic regimes introduce nonconformal behavior, Kosterlitz–Thouless transitions, and a massless phase with central charge c≈1, enriching its phase diagram.

Searching arXiv for the cited clock-model papers to verify metadata and grounding. The three-state quantum clock model is a one-dimensional $\mathbb Z_3$ lattice model whose local degrees of freedom are three-level “clock” variables acted on by clock and shift operators obeying a discrete Weyl algebra. In its standard quantum clock-chain form, it interpolates between a $\mathbb Z_3$-broken ordered phase and a disordered phase, with a second-order quantum critical point at self-duality. In related chiral and antiferromagnetic realizations, the same three-state structure supports nonconformal critical dynamics, Kosterlitz–Thouless transitions, and a massless incommensurate phase. Recent work has emphasized not only equilibrium scaling but also dynamical diagnostics based on the Loschmidt echo, rate functions, and finite-time scaling under driven protocols (Tang et al., 2023).

1. Algebraic structure and canonical Hamiltonians

The standard $q$-state quantum clock chain on an $N$-site ring with periodic boundary conditions is defined by

$$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$

where the local Hilbert space at each site is spanned by $\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$, and

$$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$

These operators satisfy

$V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$

For the three-state case, $\omega=e^{2\pi i/3}$ and, in the basis $\{|0\rangle,|1\rangle,|2\rangle\}$,

$\mathbb Z_3$0

The Hamiltonian becomes

$\mathbb Z_3$1

(Tang et al., 2023).

A closely related formulation is the three-state quantum chiral clock model, written in terms of operators $\mathbb Z_3$2 and $\mathbb Z_3$3 satisfying

$\mathbb Z_3$4

with Hamiltonian

$\mathbb Z_3$5

In the time-reversal-symmetric case considered numerically, $\mathbb Z_3$6 and the chirality parameter is $\mathbb Z_3$7, with most calculations using $\mathbb Z_3$8 (Huang et al., 2019).

In the antiferromagnetic regime, the three-state chiral clock model can also be written as

$\mathbb Z_3$9

or, at $q$0, as the mixed ferro-antiferromagnetic Potts chain

$q$1

with

$q$2

These forms make explicit that the three-state clock, Potts, and chiral-clock descriptions share the same local $q$3 operator algebra while differing in coupling structure and phase organization (Dai et al., 2016).

2. Equilibrium criticality of the standard three-state clock chain

For $q$4, the quantum $q$5-state clock model has a single second-order $q$6 quantum critical point at the self-dual coupling $q$7, with $q$8 throughout. In the three-state case, the ordered phase at $q$9 breaks the $N$0 symmetry and has three nearly degenerate clock-ordered ground states, whereas the disordered phase at $N$1 is dominated by the transverse-field term and has a unique ground state given by the uniform superposition in the $N$2-basis (Tang et al., 2023).

At the critical point $N$3, the correlation-length exponent is $N$4, while the numerical extraction reported from dynamical scaling gives $N$5. The agreement between these values identifies the three-state clock chain with the same equilibrium critical behavior as the exact three-state Potts result quoted in the dynamical study (Tang et al., 2023).

A common simplification is to treat all three-state clock models as if they shared the same critical structure. The available results distinguish several cases. The standard self-dual chain has a single second-order critical point at $N$6 for $N$7 (Tang et al., 2023). By contrast, in the chiral model with $N$8 one has the ordinary quantum three-state Potts chain with a conformal critical point of central charge $N$9, while for $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$0 the critical point is no longer conformal and the hallmark is a noninteger dynamical exponent $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$1 (Huang et al., 2019). In the antiferromagnetic region of the chiral model, the phase diagram contains commensurate, incommensurate “floating,” and disordered phases, separated by Kosterlitz–Thouless transitions (Dai et al., 2016). This suggests that the phrase “three-state quantum clock model” names a family of closely related $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$2 chains rather than a single universal phase diagram.

3. Loschmidt echo and small-quench dynamical scaling

A central dynamical result for the standard three-state clock chain is that phase transitions can be characterized by an enhanced decay behavior of the Loschmidt echo under a small quench. The protocol prepares the system in the ground state $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$3 at $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$4 and suddenly quenches to $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$5 with small $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$6. The Loschmidt echo is

$$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$7

The first minimum $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$8 at time $$H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1,$$9 obeys the finite-size scaling law

$\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$0

Equivalently,

$\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$1

(Tang et al., 2023).

The same exponent can be extracted from the short-time-averaged rate function. Defining

$\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$2

and

$\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$3

one locates the peak of $\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$4 in $\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$5 for each $\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$6, defining a pseudo-critical $\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$7, and then measures the corresponding $\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$8. This again yields

$\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$9

with numerical estimate $$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$0 (Tang et al., 2023).

These results are significant because they show that equilibrium critical exponents can be recovered from nonequilibrium observables defined through wave-function overlaps. In the three-state case, the dynamical exponent $$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$1 is not introduced as an independent quantity but as a rewriting of the correlation-length scaling through $$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$2. The method therefore ties critical spectroscopy in finite systems to Loschmidt-echo minima rather than to conventional static susceptibilities.

4. Dynamical quantum phase transitions in the three-state chain

For the three-state clock chain, dynamical quantum phase transitions are analyzed through a large quench from $$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$3 to $$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$4. In this case the Loschmidt amplitude can be written in transfer-matrix form,

$$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$5

and admits the closed expression

$$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$6

The associated rate function,

$$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$7

has its first non-analytic cusp at

$$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$8

(Tang et al., 2023).

A time-dependent $$U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}.$$9 order parameter is defined as

$V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$0

which for $V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$1 reduces exactly to

$V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$2

Its first zero occurs at the same critical time,

$V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$3

demonstrating a one-to-one correspondence between Loschmidt-rate cusps and order-parameter zeros for $V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$4 (Tang et al., 2023).

Within the same comparison across $V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$5, the first DQPT cusp obeys

$V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$6

so that the first-cusp rate function grows logarithmically with $V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$7. For $V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$8, the correspondence between singularities of the Loschmidt echo and zeros of the order parameter no longer exists; the Loschmidt echo near its first minimum converges, while the order parameter at its first zero increases linearly with $V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1.$9 (Tang et al., 2023). In the three-state case, however, the DQPT structure remains in the regime where cusp times and order-parameter zeros coincide exactly.

5. Chiral deformation and nonequilibrium finite-time scaling

The three-state quantum chiral clock model generalizes the standard chain by introducing chirality through the phase $\omega=e^{2\pi i/3}$0 on bonds. When $\omega=e^{2\pi i/3}$1, the model is the ordinary quantum three-state Potts chain and exhibits a conformal critical point with central charge $\omega=e^{2\pi i/3}$2. For $\omega=e^{2\pi i/3}$3, space-inversion and time-reversal are broken, and the critical point is no longer conformal. Field-theory and density-matrix-renormalization-group studies summarized in the finite-time-scaling work show that, for small $\omega=e^{2\pi i/3}$4, the incommensurate phase shrinks to a point and one obtains a direct, continuous transition between an ordered $\omega=e^{2\pi i/3}$5-broken phase and a disordered phase. The hallmark is a noninteger dynamical exponent $\omega=e^{2\pi i/3}$6 (Huang et al., 2019).

The order parameter is

$\omega=e^{2\pi i/3}$7

and the distance to criticality is defined by $\omega=e^{2\pi i/3}$8. Two driven protocols are considered. For a transverse-field quench,

$\omega=e^{2\pi i/3}$9

with $\{|0\rangle,|1\rangle,|2\rangle\}$0, the finite-time-scaling ansatz is

$\{|0\rangle,|1\rangle,|2\rangle\}$1

The zero-crossing and amplitude scale as

$\{|0\rangle,|1\rangle,|2\rangle\}$2

For a longitudinal-field quench,

$\{|0\rangle,|1\rangle,|2\rangle\}$3

the scaling form becomes

$\{|0\rangle,|1\rangle,|2\rangle\}$4

with

$\{|0\rangle,|1\rangle,|2\rangle\}$5

(Huang et al., 2019).

All simulations use infinite-MPS time evolution (iTEBD), with bond dimension up to $\{|0\rangle,|1\rangle,|2\rangle\}$6. For $\{|0\rangle,|1\rangle,|2\rangle\}$7, the extracted critical parameters are

$\{|0\rangle,|1\rangle,|2\rangle\}$8

The best collapse of the two-point correlation function

$\{|0\rangle,|1\rangle,|2\rangle\}$9

with

$\mathbb Z_3$00

occurs at $\mathbb Z_3$01 (Huang et al., 2019).

The same work extends finite-time scaling to thermal initial states. For the transverse protocol,

$\mathbb Z_3$02

and analogously for the longitudinal protocol,

$\mathbb Z_3$03

Perfect collapse is reported even when starting from a thermal state close to criticality, confirming that finite temperature can be treated as an additional scaling field of dimension $\mathbb Z_3$04 (Huang et al., 2019).

6. Antiferromagnetic regime, massless phase, and entanglement diagnostics

In the antiferromagnetic region of the three-state quantum chiral clock model, the $\mathbb Z_3$05 phase diagram contains three phases: a commensurate ordered phase, an incommensurate floating phase, and a disordered phase. Two Kosterlitz–Thouless transitions occur for $\mathbb Z_3$06. At $\mathbb Z_3$07, the transition between disordered and incommensurate phases is

$\mathbb Z_3$08

inferred from the Potts-chain critical field

$\mathbb Z_3$09

At $\mathbb Z_3$10, the exact duality $\mathbb Z_3$11, $\mathbb Z_3$12 gives

$\mathbb Z_3$13

(Dai et al., 2016).

The critical point $\mathbb Z_3$14 is obtained from the bipartite von Neumann entropy of an infinite matrix-product state with bond dimension $\mathbb Z_3$15,

$\mathbb Z_3$16

where $\mathbb Z_3$17 are Schmidt coefficients. The entropy exhibits a pronounced peak at a pseudo-critical field $\mathbb Z_3$18, and extrapolating

$\mathbb Z_3$19

gives the quoted critical value (Dai et al., 2016).

Within the massless phase $\mathbb Z_3$20, finite-entanglement scaling yields

$\mathbb Z_3$21

and the central charge is estimated as

$\mathbb Z_3$22

At the special point $\mathbb Z_3$23, corresponding to the purely antiferromagnetic three-state Potts model, one recovers exactly $\mathbb Z_3$24. The equivalent finite-block formula is

$\mathbb Z_3$25

(Dai et al., 2016).

The spin-spin correlator

$\mathbb Z_3$26

decays algebraically in the massless phase,

$\mathbb Z_3$27

with continuously varying exponent $\mathbb Z_3$28. Numerically, one finds $\mathbb Z_3$29 at $\mathbb Z_3$30, while as $\mathbb Z_3$31 approaches $\mathbb Z_3$32, $\mathbb Z_3$33 decreases continuously and reaches approximately $\mathbb Z_3$34 at $\mathbb Z_3$35. For $\mathbb Z_3$36, the correlation decays exponentially, $\mathbb Z_3$37 (Dai et al., 2016). These results distinguish the antiferromagnetic chiral-clock regime from the direct ordered-to-disordered transition emphasized in the small-chirality nonequilibrium study.

7. Representative results and conceptual distinctions

The following values summarize the principal three-state results reported in the cited studies.

Regime	Quantity	Value
Standard clock chain	Critical point	$\mathbb Z_3$38
Standard clock chain	Correlation-length exponent	$\mathbb Z_3$39
Standard clock chain	Small-quench exponent	$\mathbb Z_3$40
Standard clock chain	First DQPT time	$\mathbb Z_3$41
Standard clock chain	First-cusp rate value	$\mathbb Z_3$42
Chiral clock model, $\mathbb Z_3$43	Critical point	$\mathbb Z_3$44
Chiral clock model, $\mathbb Z_3$45	Dynamical exponent	$\mathbb Z_3$46
Antiferromagnetic Potts line	Critical field	$\mathbb Z_3$47
Massless antiferromagnetic phase	Central charge	$\mathbb Z_3$48

Several conceptual distinctions follow directly from these results. First, the standard three-state clock chain is self-dual and exhibits a single second-order critical point at $\mathbb Z_3$49 (Tang et al., 2023). Second, adding chirality changes the critical dynamics qualitatively: for small $\mathbb Z_3$50, the transition remains direct and continuous but becomes nonconformal, with noninteger $\mathbb Z_3$51 (Huang et al., 2019). Third, in the antiferromagnetic region the phase diagram is not reduced to a single critical point; instead it includes a massless incommensurate phase bounded by Kosterlitz–Thouless transitions and characterized by $\mathbb Z_3$52 and continuously varying $\mathbb Z_3$53 (Dai et al., 2016).

A common misconception is that dynamical signatures in $\mathbb Z_3$54 chains are universally equivalent across all parameter regimes. The available evidence supports a more specific statement: in the standard $\mathbb Z_3$55 clock chain, large quenches produce a one-to-one correspondence between Loschmidt-rate cusps and zeros of the order parameter, and the first cusp satisfies $\mathbb Z_3$56 (Tang et al., 2023). This correspondence was established for the standard clock chain for $\mathbb Z_3$57; it should not be transferred without qualification to the chiral or antiferromagnetic regimes, where the cited works focus instead on finite-time scaling, entanglement entropy, and massless-phase diagnostics.

Taken together, these results place the three-state quantum clock model at the intersection of discrete-symmetry breaking, nonconformal quantum criticality, and nonequilibrium scaling. The standard chain provides a controlled setting in which equilibrium exponents can be extracted from Loschmidt-echo scaling and DQPT cusps, while the chiral and antiferromagnetic variants demonstrate that the same local $\mathbb Z_3$58 algebra supports qualitatively different critical structures, including direct continuous transitions with noninteger $\mathbb Z_3$59 and extended $\mathbb Z_3$60 massless phases (Tang et al., 2023).